91Ó°ÊÓ

A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is an imaginary unit with the property that \(i^2 = -1\). The complex number combines both real and imaginary parts:

• Real part: \(a\)
• Imaginary part: \(bi\)

Complex numbers are used in a variety of fields such as engineering, physics, and applied mathematics. They allow for solutions to equations that cannot be solved using only real numbers. For instance, the equation \(x^2 + 1 = 0\) has no real solution but has two complex solutions: \(i\) and \(-i\).

The modulus (or absolute value) of a complex number \(z = a + bi\) is denoted as \(|z|\) and measures the distance of the complex number from the origin in the complex plane. Mathematically, it is given by \[|z| = \sqrt{a^2 + b^2}\.\]

Some important properties of the modulus include:

• It is always a non-negative real number.
• For any two complex numbers \(z_1\) and \(z_2\), the modulus of their product can be expressed as \( |z_1 z_2| = |z_1| |z_2| \).
• The modulus of a complex number remains the same whether the number is written in Cartesian form \(a + bi\) or in polar form \(r(cos(\theta) + i sin(\theta))\).

For example, if you have a complex number \(z = 3 + 4i\), its modulus would be \[|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5.\]

Multiplying complex numbers involves both their real and imaginary parts. To multiply \(z_1 = a + bi\) and \(z_2 = c + di\), you use the distributive property:

\begin{aligned} z_1 \times z_2 & = (a + bi)(c + di) \ &= ac + adi + bci + bdi^2\ \ &= ac + adi + bci - bd \ \ &= (ac - bd) + (ad + bc)i \ \text{(since } \(i^2 = -1\)) \end{aligned}

This yields a new complex number where:

• The real part is \(ac - bd\),
• The imaginary part is \(ad + bc\).

In general, when dealing with the modulus in multiplications, using the property \( |z_1 z_2| = |z_1| |z_2| \) can greatly simplify calculations, as seen in the given exercise.

To understand why \(|z^2| = |z|^2\), note that \( z^2 = z \times z \) follows the multiplication property of modulus: \[ |z^2| = |z \times z| = |z| \times |z| = |z|^2. \]